Substitution systems and the three versions of distributional chaos

Wang, Hui; Liao, Gongfu; Qin-jie, Fan

doi:10.1016/j.topol.2008.07.021

Cited by 13 publications

(6 citation statements)

References 9 publications

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“…We tried too, but even though we were able to construct an infinite DC3-set, it was not bigger than countable. Fortunately, just when we were ready to give up, we found out that there is an example of a subshift of the full 5-shift in [14], which has the required proprerties. And so we just need to construct a Gehman-like dendrite for the full 5-shift and its subshifts.…”

Section: Uncountable Dc3-setmentioning

confidence: 99%

Distributional Chaos and Dendrites

Roth

2018

Int. J. Bifurcation Chaos

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Many definitions of chaos have appeared in the last decades and with them the question if they are equivalent in some more specific spaces. Our focus will be distributional chaos, first defined in 1994 and later subdivided into three major types (and even more subtypes). These versions of chaos are equivalent on a closed interval, but distinct in more complicated spaces. Since dendrites have much in common with the interval, we explore whether or not we can distinguish these kinds of chaos already on dendrites. In the end of the paper we will also briefly look at the correlation with other types of chaos.

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Section: Uncountable Dc3-setmentioning

confidence: 99%

Distributional Chaos and Dendrites

Roth

2018

Int. J. Bifurcation Chaos

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“…Thus only the substitution system of non-primitive constant-length substitution can have uncountable DC scrambled set. Such example is given in [13].…”

Section: Corollary 34 If (X θ σ ) Has DC Pairs Then a ≥mentioning

confidence: 99%

Distributional chaos in constant-length substitution systems

Wang

Qin-jie

Liao

2010

Nonlinear Analysis: Theory, Methods & Applications

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“…By the definition we immediately have that DC1 implies DC2 and DC2 implies DC3, and it is also known that none of the reverse implications holds (e.g., see [4]). It can also be proved that DC1 or DC2 implies chaos in the sense of Li and Yorke, but DC3 does not [5].…”

Section: Introductionmentioning

confidence: 97%